Patterns and Sequences
Describe a sequence by its position-to-term rule rather than by what came before it, and use the nth-term formula to jump straight to any term.
Before diving into the practice questions, take a moment to think about what you already know about this topic.
Know
- Arithmetic sequences have a fixed common difference $d$ between consecutive terms
- $d$ can be found by calculating $T_2 - T_1$
- The first term is labelled $a$
Understand
- Each term can be expressed using the nth-term formula $T_n = a + (n-1)d$
- This formula always simplifies to a linear expression of the form $T_n = dn + c$
- If substituting a value gives a non-integer $n$, that value is not in the sequence
Can Do
- Find the common difference of any arithmetic sequence
- Write the nth-term rule for a given sequence
- Find any specific term or test whether a value belongs to a sequence
A sequence is an ordered list of numbers following a rule. In an arithmetic sequence, you add (or subtract) the same amount each time, this fixed amount is called the common difference $d$.
Instead of listing every term, we use a formula that gives any term directly. For an arithmetic sequence with first term $a$ and common difference $d$:
$$T_n = a + (n-1)d$$
To use it: substitute $a$, $d$, and the term number $n$. Then expand and simplify to get a clean linear expression.
Text description
- The formula Tₙ = a + (n − 1)d in a box.
- A red line drops from the a to the label first term.
- A teal line drops from the d to the label common difference.
Example: Find the nth-term formula for 3, 7, 11, 15...
- $a = 3$, $d = 4$
- $T_n = 3 + (n-1) \times 4 = 3 + 4n - 4 = 4n - 1$
Check: $T_1 = 4(1) - 1 = 3$ ✓ $T_4 = 4(4) - 1 = 15$ ✓
The simplified form is always $T_n = dn + c$, a linear expression in $n$.
Once you have the nth-term formula, you can find any term instantly, or test whether a number belongs to the sequence.
To generate a number pattern from an algebraic expression, substitute consecutive whole-number positions. For $T_n=3n+2$, using $n=1,2,3,4$ gives $5,8,11,14$, so the expression generates a pattern increasing by 3.
$a = 5$, $d = 4$ → $T_n = 4n + 1$
$T_{20} = 4(20) + 1 = 80 + 1 = 81$
Set $4n - 1 = 45$ → $4n = 46$ → $n = \frac{46}{4} = 11.5$, not a whole number, so 45 is NOT in this sequence.
Watch Me Solve It · Worked example
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1Find the common difference$d = 5 - 2 = 3$Subtract any term from the next. Check: $8 - 5 = 3$ and $11 - 8 = 3$. Consistent, so this is arithmetic with $d = 3$.
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2Write the formula and simplify$T_n = 2 + (n-1) \times 3 = 2 + 3n - 3 = 3n - 1$Substitute $a = 2$ and $d = 3$ into $T_n = a + (n-1)d$. Expand the bracket, then collect the constants: $2 - 3 = -1$.
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3Find $T_{15}$$T_{15} = 3(15) - 1 = 45 - 1 = 44$Substitute $n = 15$ into the simplified formula. Multiply first, then subtract.
Quick Check · 5 questions
Q1. Find the term-to-term rule and the next two terms of: 15, 11, 7, 3, -1...
Q2. Find the $n$th term of the sequence: 6, 10, 14, 18, 22...
Q3. The $n$th term of a sequence is $6n - 5$.
Which numbers belong to a sequence?
Which numbers belong to a sequence?. A growing pattern has 3, 7, 11 and 15 tiles in its first four stages. Each new stage adds 4 tiles. Stage numbers start at $n=1$.
Write a rule for the number of tiles at stage $n$. Find the 21st stage. Decide whether 99 and 100 tiles can occur, and justify each decision.
Show your working first, then open the worked solution to check your reasoning.
Check the worked solution
The rule is $T=4n-1$: it gives 3 at $n=1$ and increases by 4 each stage. At stage 21, $T=4(21)-1=83$. For 99 tiles, $4n-1=99$ gives $n=25$, so 99 occurs. For 100 tiles, $n=101/4=25.25$, which is not a positive whole-number stage. Therefore 100 does not occur.
Key Concept
Review the main ideas from this lesson.
Formulas
Key formulas and rules.
Watch Out
Common mistakes to avoid.
Check
Always verify your answers.
Practice
Keep practicing to master.
Next
Build on these skills.
Interactive: Algebra Machine
Substitute numbers into algebraic expressions and see them evaluate step by step.
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Brain Trainer
Set a timer for 3 minutes. Find the $n$th term for each sequence.